Subgroups — C₃×C₅×#2255 #14275289

Order 1920.

240 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#499	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#367	C₂	1	2	yes	maximal
4	C₃×C₅×#366	C₂	1	2	yes	maximal
5	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic
6	C₅×#2255	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
8	C₃×C₅×#125	—	2	1	no	—
9	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
10	C₃×C₅×#107	—	2	1	no	—
11	C₃×C₅×#107	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	C₂²	1	1	yes	characteristic
13	C₃×C₅×#107	C₂²	1	2	yes	—
14	C₃×C₅×#106	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#2255	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#500	C₂×C₃	1	1	yes	characteristic
18	C₅×#499	C₂×C₃	1	1	yes	characteristic
19	C₅×#367	C₂×C₃	1	2	yes	—
20	C₅×#366	C₂×C₃	1	2	yes	—
21	C₅×#346	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₅×Q₈	—	4	1	no	—
23	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₃×C₅×C₄⋊C₄	—	2	1	no	—
26	C₃×C₅×C₄⋊C₄	—	2	1	no	—
27	C₃×C₅×C₄⋊C₄	—	2	2	no	—
28	C₃×C₅×C₄⋊C₄	—	4	1	no	—
29	C₃×C₅×C₄⋊C₄	—	4	1	no	—
30	C₃×C₅×C₄⋊C₄	—	4	1	no	—
31	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×C₁₆	Q₈	1	4	yes	complemented
33	C₃×#500	C₂×C₅	1	1	yes	characteristic
34	C₃×#499	C₂×C₅	1	1	yes	characteristic
35	C₃×#367	C₂×C₅	1	2	yes	—
36	C₃×#366	C₂×C₅	1	2	yes	—
37	C₃×#346	C₂×C₅	1	1	yes	characteristic
38	C₅×C₄⋊Q₈	—	2	1	no	—
39	C₅×#125	—	2	1	no	—
40	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
41	C₅×#107	—	2	1	no	—
42	C₅×#107	C₂²×C₃	1	1	yes	characteristic
43	C₅×#107	C₂²×C₃	1	2	yes	—
44	C₅×#107	C₂²×C₃	1	1	yes	characteristic
45	C₅×#106	—	2	1	no	—
46	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	#2255	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₃×C₅×Q₈	—	8	2	no	—
49	C₂×C₃×C₄×C₅	—	4	1	no	—
50	C₂×C₃×C₄×C₅	—	4	1	no	—
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₃×C₅×C₈	Q₁₆	1	2	yes	—
56	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
57	C₃×C₄⋊Q₈	—	2	1	no	—
58	C₃×#125	—	2	1	no	—
59	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
60	C₃×#107	—	2	1	no	—
61	C₃×#107	C₂²×C₅	1	1	yes	characteristic
62	C₃×#107	C₂²×C₅	1	2	yes	—
63	C₃×#107	C₂²×C₅	1	1	yes	characteristic
64	C₃×#106	—	2	1	no	—
65	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
66	C₂×C₅×Q₈	—	4	1	no	—
67	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
69	C₅×C₄⋊C₄	—	2	1	no	—
70	C₅×C₄⋊C₄	—	2	1	no	—
71	C₅×C₄⋊C₄	—	2	2	no	—
72	C₅×C₄⋊C₄	—	4	1	no	—
73	C₅×C₄⋊C₄	—	4	1	no	—
74	C₅×C₄⋊C₄	—	4	1	no	—
75	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
76	C₅×C₁₆	C₃×Q₈	1	4	yes	complemented
77	#500	C₂×C₃×C₅	1	1	yes	characteristic
78	#499	C₂×C₃×C₅	1	1	yes	characteristic
79	#367	C₂×C₃×C₅	1	2	yes	—
80	#366	C₂×C₃×C₅	1	2	yes	—
81	#346	C₂×C₃×C₅	1	1	yes	characteristic
82	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
83	C₃×C₄×C₅	—	2	1	no	—
84	C₃×C₄×C₅	—	2	1	no	—
85	C₃×C₄×C₅	—	8	1	no	—
86	C₃×C₄×C₅	—	8	2	no	—
87	C₃×C₄×C₅	—	8	1	no	—
88	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
89	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₃×Q₈	—	4	1	no	—
91	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
93	C₃×C₄⋊C₄	—	2	1	no	—
94	C₃×C₄⋊C₄	—	2	1	no	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	—	4	1	no	—
97	C₃×C₄⋊C₄	—	4	1	no	—
98	C₃×C₄⋊C₄	—	4	1	no	—
99	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
100	C₃×C₁₆	C₅×Q₈	1	4	yes	complemented
101	C₅×Q₈	—	8	2	no	—
102	C₂×C₄×C₅	—	4	1	no	—
103	C₂×C₄×C₅	—	4	1	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
108	C₅×C₈	C₃×Q₁₆	1	2	yes	—
109	C₅×C₈	C₂×C₃×Q₈	1	2	yes	—
110	C₄⋊Q₈	—	2	1	no	—
111	#125	—	2	1	no	—
112	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
113	#107	—	2	1	no	—
114	#107	C₂²×C₃×C₅	1	1	yes	characteristic
115	#107	C₂²×C₃×C₅	1	2	yes	—
116	#107	C₂²×C₃×C₅	1	1	yes	characteristic
117	#106	—	2	1	no	—
118	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
119	C₂×C₃×C₅	#500	1	1	yes	characteristic
120	C₂×C₃×C₅	#509	1	1	yes	characteristic
121	C₂×C₃×C₅	#510	1	1	yes	characteristic
122	C₃×Q₈	—	8	2	no	—
123	C₂×C₃×C₄	—	4	1	no	—
124	C₂×C₃×C₄	—	4	1	no	—
125	C₂×C₃×C₄	—	4	2	no	—
126	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
127	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
129	C₃×C₈	C₅×Q₁₆	1	2	yes	—
130	C₃×C₈	C₂×C₅×Q₈	1	2	yes	—
131	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
132	C₄×C₅	—	2	1	no	—
133	C₄×C₅	—	2	1	no	—
134	C₄×C₅	—	8	1	no	—
135	C₄×C₅	—	8	2	no	—
136	C₄×C₅	—	8	1	no	—
137	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
138	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
139	C₂×Q₈	—	4	1	no	—
140	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
141	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
149	C₁₆	C₃×C₅×Q₈	1	4	yes	complemented
150	C₃×C₅	#2255	1	1	yes	characteristic, direct product factor, Hall
151	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
152	C₃×C₄	—	2	1	no	—
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	8	1	no	—
155	C₃×C₄	—	8	2	no	—
156	C₃×C₄	—	8	1	no	—
157	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
158	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
159	C₂×C₅	C₃×#500	1	1	yes	characteristic
160	C₂×C₅	C₃×#509	1	1	yes	characteristic
161	C₂×C₅	C₃×#510	1	1	yes	characteristic
162	Q₈	—	8	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
167	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
168	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
169	C₈	C₃×C₅×Q₁₆	1	2	yes	—
170	C₈	C₂×C₃×C₅×Q₈	1	2	yes	—
171	C₂×C₃	C₅×#500	1	1	yes	characteristic
172	C₂×C₃	C₅×#509	1	1	yes	characteristic
173	C₂×C₃	C₅×#510	1	1	yes	characteristic
174	C₅	C₃×#2255	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	2	no	—
179	C₄	—	8	1	no	—
180	C₄	—	8	1	no	—
181	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
182	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
183	C₃	C₅×#2255	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	C₃×C₅×#500	1	1	yes	characteristic
185	C₂	C₃×C₅×#509	1	1	yes	characteristic
186	C₂	C₃×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
6. C₅×#2255	C₃	direct (N × H)
16. C₃×#2255	C₅	direct (N × H)
32. C₃×C₅×C₁₆	Q₈	semidirect
47. #2255	C₃×C₅	direct (N × H)
76. C₅×C₁₆	C₃×Q₈	semidirect
100. C₃×C₁₆	C₅×Q₈	semidirect
149. C₁₆	C₃×C₅×Q₈	semidirect
150. C₃×C₅	#2255	direct (N × H)
174. C₅	C₃×#2255	direct (N × H)
183. C₃	C₅×#2255	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#500	2
C₃×C₅×#499	2
C₃×C₅×#367	2
C₃×C₅×#366	2
C₃×C₅×#346	2
C₅×#2255	3
C₃×#2255	5

Subgroups — C3×C5×#2255 #14275289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2255 #14275289